Let $X, Y$ be independent random variables with values in $(0, \infty)$ and the same probability density $\frac{2}{\sqrt{\pi}} e^{-x^{2}}$. Let $U=X^{2}+Y^{2}, V=Y / X$. Compute the joint probability density of $U, V$ and the marginal densities of $U$ and $V$ respectively. Are $U$ and $V$ independent?